Bose-Einstein Condensates in Rotating Traps and Optical ... - BEC

172 Condensate fraction
confirming that the system is coherent as long as EJ/EC ≪ 1 (see section 10.1).
When the exponent of the power law takes the value ν =0.14, corresponding to EJ =
1.62EC, the 1D system is expected to exhibit the Bradley-Doniach phase transition to an
insulating phase where the 1-body density matrix decays exponentially [157]. Note however
that before this transition is reached the depletion (12.72) becomes large and hence Bogoliubov
theory is no longer applicable.
The transition from the regime where (12.62) is applicable to the regime where the 1D
character of the fluctuations prevail (see Eq.(12.72) can be estimated by calculating at what
lattice depth s the condition
ã 2Nw
≈ ν ln , (12.78)
d π
is fulfilled. As an example, for gn =0.2ER, Nw = 200 and N = 500, this transition is
predicted to occur around s =30where the left and right side of the inequality become equal
to ∼ 4%.
Quantum depletion in current experiments
To give an example, we set gn =0.2ER,N = 200 and Nw = 500 describing a setting similar
to the experiment of [73]. Bogoliubov theory predicts a depletion of ≈ 0.6% in the absence
of the lattice (s =0). At a lattice depth of s =10the evaluation of Eq.(12.1), using the
tight binding result (12.57), and keeping the sum discrete yields a depletion of ≈ 1.7%. On
the other hand, Eq.(12.60), obtained by replacing the sum in Eq.(12.57) by an integral, yields
a depletion of ≈ 2%, in reasonable agreement with the full result ≈ 1.7%. The 2D formula
(12.62) instead yields ≈ 2.9% depletion, revealing that the system is not yet fully governed by
2D fluctuations. With the same choice of parameters, the power law exponent (12.73) has the
value ν =0.001 and the 1D depletion (12.72) is predicted to be ≈ 0.6%, significantly smaller
than the full value ≈ 1.7%. This reveals that the sum (12.57) is not exhausted by the terms
with px = py =0. In conclusion, one finds that for this particular setting, the character of
fluctuations is intermediate between 3D and 2D, and still far from 1D. In particular, from the
above estimates it emerges that in order to reach the conditions for observing the Bradley-
Doniach transition one should work at much larger values of s.
A very different situation is encountered in the experiment [91]: There, a weak onedimensional
optical lattice of depth sax is set up along the tubes formed by a deep twodimensional
lattice. In this setting, the number of particles per site is very small N ≈ 3 − 4.
For sax =4and interaction strength gn =1ER one finds Nν =0.42, yielding a large depletion
of 0.55 for Nw =40sites and N =3particles per site. Hence, this setting prepares the
gas in a regime of strong coupling beyond the validity of GP- and Bogoliubov theory. Note
that in determining the interaction parameter gn we have taken into account the non-uniform
confinement in the transverse directions.